A Scalar Optimization Proof of Sendov's Conjecture with Reduced Computer Assistance

Abstract

Let $p$ be a complex polynomial of degree $n\ge2$ whose zeros lie in the closed unit disk. Sendov's conjecture asserts that every zero of $p$ lies within distance one of a zero of $p'$. Mazur's recent proof, and Tao's streamlined exposition of it, reduce a hypothetical counterexample to a scalar lower bound \[ 1\le F(\eta,\alpha,n) \] together with two upper bounds for the endpoint parameter $\eta$ and the uniform restriction $0<\alpha\le 17$. We complete this scalar reduction by a two-stage optimization argument. First, $F$ is nondecreasing in $\eta$, so $\eta$ may be replaced by a piecewise polar envelope $\eta^{\ast}(\alpha)$. Writing $s=(n-1)/2$ converts the degree to a half-integer variable and gives \[ F(\eta^{\ast}(\alpha),\alpha,n) = E(\alpha,s) + \kappa(\alpha,s) \int_0^1 t^3 \widehat{\beta}(t;\alpha,s)^{\,s-\frac{3}{2}} \,dt. \] On each half-unit $\alpha$-slab, $E$ and $\kappa$ decrease with $\alpha$, whereas $\widehat\beta$ increases. Convexity in $t$ reduces all but the first mesh interval to eight explicit nodal terms. The first interval satisfies a uniform bound $3/200$. Each nodal upper term is a strictly log-concave function of the half-integer $s$, so its global discrete maximum is certified by two adjacent ratio evaluations. A fixed finite certificate over the $34$ half-unit slabs gives \[ F(\eta^{\ast}(\alpha),\alpha,n)<\frac{97}{100}<1, \] contradicting the scalar lower bound. The main contribution is a substantial reduction of the computer assistance required after the Mazur--Tao scalar reduction: all continuous optimization and the unbounded degree parameter are handled analytically, leaving only a small fixed collection of explicit one-variable inequalities.